Question

Magnetic field $\overrightarrow{B}=3x\hat{i}$ exists in space. Energy stored in a cylinderical region of length 1m along x-axis with one end at origin is [Radius of the cylinder is 2m]

• A. $1.5\times {10}^{-7}J$
• B. $2\times {10}^{6}J$
• C. $1.5\times {10}^{7}J$
• D. $1.6\times {10}^{-6}J$

Answer 1

The correct option is C $1.5\times {10}^{7}J$

Say $dE$ is the energy present in the small volume $dv$. Then,
$dE=\frac{B^2}{2{\mu }_0}dv$
$dE=\frac{B^2}{2{\mu }_0}\pi R^{2}dx$
$dE=\frac{9x^2}{2{\mu }_0}\pi R^{2}dx$
$E=\int dE=\frac{9\pi R^2}{2{\mu }_0}{\int }_0^1{x}^{2}dx=\frac{9\pi R^2}{2{\mu }_0}.\frac{1}{3}=1.5\times {10}^{7}J$